SOLUTION: The sum of the first two terms of a geometric series is 90. The third term is 24. A. Show That there are two possible series and find the term and the common ration in each case

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Question 1092202: The sum of the first two terms of a geometric series is 90. The third term is 24.
A. Show That there are two possible series and find the term and the common ration in each case.
B. Show that both series converge and find their respective sums.
Answer by greenestamps(13327) About Me  (Show Source):
You can put this solution on YOUR website!


The third term is 24:

ar%5E2=24
--> a+=+24%2Fr%5E2 (1)

The sum of the first two terms is 90:

a%2Bar+=+a%281%2Br%29+=+90 (2)

Substitute (1) into (2):

%2824%2Fr%5E2%29%281%2Br%29+=+90
24%281%2Br%29+=+90r%5E2
90r%5E2-24r-24+=+0
15r%5E2-4r-4+=+0
%285r%2B2%29%283r-2%29+=+0

The two values of the common ratio are -2/5 and 2/3.

For the common ratio -2/5, the first term a is

24%2F%28-2%2F5%29%5E2+=+24%2F%284%2F25%29+=+150

That gives us the sequence 150, -60, 24, ...

For the common ratio 2/3, the first term a is

24%2F%282%2F3%29%5E2+=+24%2F%284%2F9%29+=+54

That gives us the sequence 54, 36, 24, ...